To complete the square for the given equation [tex]\( x^2 + 12x = 11 \)[/tex], follow these steps:
1. Identify the coefficient of [tex]\( x \)[/tex] (which we'll call [tex]\( b \)[/tex]):
The given equation is [tex]\( x^2 + 12x = 11 \)[/tex]. Here, the coefficient [tex]\( b \)[/tex] is 12.
2. Calculate [tex]\(\left(\frac{b}{2}\right)^2\)[/tex]:
Divide the coefficient [tex]\( b \)[/tex] by 2 and then square the result.
[tex]\[ b = 12 \][/tex]
[tex]\[ \frac{b}{2} = \frac{12}{2} = 6 \][/tex]
[tex]\[ \left(\frac{b}{2}\right)^2 = 6^2 = 36 \][/tex]
3. Add this value to both sides of the original equation to complete the square:
The value that should be added to both sides is 36.
Therefore, the number that should be added to both sides of the equation [tex]\( x^2 + 12x = 11 \)[/tex] to complete the square is [tex]\( \boxed{36} \)[/tex].
